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Non-Perturbative Approach to the Landau Gauge Gluodynamics A.Y. Lokhova, Ph. Boucaudb, J.P. Leroyb, A. Le Yaouancb, J. Michelib, O. Pèneb, J. Rodríguez-Quinteroc and C. Roiesnela a Centre de Physique Théorique1de l’Ecole Polytechnique 6 F91128 Palaiseau cedex, France 0 bLaboratoire de Physique Théorique et Hautes Energies2 0 2 Université de Paris XI, Bâtiment 211, 91405 Orsay Cedex, France n c Dpto. Física Aplicada, Fac. Ciencias Experimentales, a Universidad de Huelva, 21071 Huelva, Spain. J 5 Abstract 1 v We discuss a non-perturbative lattice calculation of the ghost and gluon propaga- 7 tors in the pure Yang-Mills theory in Landau gauge. The ultraviolet behaviour is 0 checked up to NNNLO yielding the value Λnf=0 = 269(5)+12 MeV, and we show 0 MS −9 1 that lattice Green functions satisfy the complete Schwinger-Dyson equation for the 0 ghost propagator for all considered momenta. The study of the above propagators 6 0 at small momenta showed that the infrared divergence of the ghost propagator is / enhanced, whereas the gluon propagator seem to remain finite and non-zero. The t a result for the ghost propagator is consistent with the analysis of the Slavnov-Taylor l - identity, whereas, according to this analysis, the gluon propagator should diverge p e in the infrared, a result at odds with other approaches. h : v CPHT-PC076.1205, LPT-06-02, UHU-FT-18 i X r a 1 Introduction In this talk we report on a non-perturbative lattice study of basic correlation functions of the Euclidean Landau gauge pure Yang-Mills gauge theories, i.e. the gluon propagator p p G(p) G(2)ab(p,−p) = δab δ − µ ν , (1) µν µν p2 p2 (cid:18) (cid:19) and the ghost propagator F(p) F(2)ab(p,−p) = δab . (2) p2 1Unité Mixte de Recherche 7644 du Centre National de la Recherche Scientifique 2Unité Mixte de Recherche 8627 du Centre National de la Recherche Scientifique 1 These propagators have been successfully checked by perturbation theory up to NNNLO in the ultraviolet domain,([1],[2]) yielding the non-perturbative value of Λnf=0 = 269(5)+12 MeV. (3) MS −9 Now we concentrate on the infrared exponents α and α that describe power-law devi- F G ations from free propagators when p → 0 G(p2) ∝ (p2)αG F(p2) ∝ (p2)αF. (4) 2 Non-perturbative study of the infrared exponents α ,α F G Diverse analytical approaches (study of truncated Schwinger-Dyson equations and of renormalisation group equation [3], [4]) agree that the infrared divergence of the ghost propagator is enhanced, i.e. α < 0; while they predict different values for α , mostly F G around α ≈ 1.2. This means that the gluon propagator is suppressed in the infrared. G Lattice simulations confirmed the prediction for the ghost propagator, whereas the lattice gluon propagator seems to remain finite and non-zero in the infrared, i.e. α = 1 [5]. G Let us consider the Slavnov-Taylor identity [6] relating the three-gluon vertex Γ , λµν the ghost-gluon vertex Γ (p,q;r) 3 and the propagators: λµ F(p2) F(p2) pλΓ (p,q,r) = e(δ r2 −r r )Γ (r,p;q)− (δ q2 −q q )Γ (q,p;r). (5) λµν λν λ ν λµ λµ λ µ λν G(r2) G(q2) Taking the limit r → 0 keeping q andep finite, and using G(r2) ≃ (r2)αeG, one finds the following limits on the infrared exponents [7] α < 1 gluon propagator diverges in the infrared, and G (6) α ≤ 0 the divergence of the ghost propagator is enhanced in the infrared. F (cid:26) The limit on α disagrees with many other analytical predictions [3], except some cases G in [4]. Let us try to understand this discrepancy. All these methods rely on a commonly accepted relation between the infrared exponents 2α +α = 0. (7) F G which we shall discus now. The origin of this relation is the dimensional analysis of the Schwinger-Dyson equation for the ghost propagator(SD): 1 d4q F(q2)G((q−k)2) (k ·q)2 −k2q2 = 1+g2N H (q,k) , (8) F(k) 0 c (2π)4 q2(q −k)2 k2(q −k)2 1 ! Z (cid:20) (cid:21) where H (q,k) is one of the scalar functions defining the ghost-gluon vertex: 1 qν′Γν′ν(−q,k;q −k) = qνH1(q,k)+(q −k)νH2(q,k). (9) 3r is the momentum of the gluon, q is the momentum of the entering ghost. e 2 The large momentum behaviour([8],[6]) of this vertex depends on the kinematic configu- ration: pµpν ·ΓMS(−p,0;p) = 1 to all orders p2 µν (10) pµpν ·ΓMS(−p,p;0) = 1+ 9 α2(p2)+... p2 µν 16π s e Note that in the case of the vanishing momentum of the out-going ghost (and only in this e case) one has H (q,0)+H (q,0) = 1. (11) 1 2 If both H are non-singular then one can suppose H (q,k) ≃ 1 in (8), and (7) is straight- 1,2 1 forward by a dimensional analysis. However, we have a priori no reason to think that the scalar functions H (q,k) and H (q,k) are separately non-singular for all q,k. Writing, for 1 2 example, H (q,k) ∼ (q2)αΓ h q·k, k2 , with a non-singular function h , we keep all the 1 1 q2 q2 1 generality of the argument adm(cid:16)itting a(cid:17)singular behaviour of the scalar factor H (q,k). 1 Doing the dimensional analysis of eq. (8) without putting H (q,k) ≃ 1, we obtain that 1 the relation (7) is satisfied if and only if the following triple condition is verified [7]: α 6= 0 F 2α +α = 0 ⇐⇒ α = 0 (12) F G Γ  α +α < 1  F G The first and the last conditions are compatiblewith limits coming from the analysis of the Slavnon-Taylor identity (6), and are also consistent with our lattice simulations [7]. If one of the conditions (12) is not verified then (7) should be replaced by 2α +α +α = 0. (13) F G Γ In the following section we present the results of a numerical test of the relation (7), and thus we probe the validity of the condition on α . Γ 3 Lattice check of analytical predictions As we have already mentioned, lattice Green functions are checked at high precision with perturbationtheory at largemomentum. In order to test the validity of lattice predictions at small momenta we verified that lattice Green functions satisfy the ghost SD equation written in the form p F(p2) = 1+g µ fabchAc(0)·F(2)ba(A,p)i, (14) 0N2 −1 µ 1conf c e e where F(2)ba(A,p) is the correlator of the ghost and anti-ghost fields in a background field 1conf A. Note that the three-point function in the r.h.s is in a mixed coordinate-momentum represeentation, and that here we make no assumption about the vertex. The traditional form (8) differs from (14) by a Legendre transformation. We see from Fig.1 that the lattice Green functions match pretty well the SD equation (14) in both the ultraviolet and infrared regions. Now we try to check the validity of the non-singularity assumption for the scalar factor H (q,k). For this purpose we check numerically whether our lattice 1 propagators satisfy the equation (8) completed with the assumption H (q,k) = 1 (cf. 1 Fig.2 [7]). We see that at small momenta (below ≈ 3 GeV) the SD with the assumption 3 Figure 1: Checking that lattice Green functions satisfy the ghost SD equation (14). The l.h.s vs r.h.s of (14) is plotted at the left, and at the right we plot F(p2)−g pµ fabchAc(0).F(2)ba(A,p)i compared 0N2−1 µ 1conf to 1 c e e 0 -0.1 -0.2 -0.3 1/F(p2 ) - ∆~ (p2) /p2 U Loop integral -0.4 -0.5 0 0.5 1 1.5 2 ap Figure2: CheckingwhetherlatticeGreenfunctionssatisfytheghostSDequation(8)withanassumption H (q,k)=1. The upper line(circles)correspondto the loopintegralin(8), andthe downline (triangles) 1 corresponds to 1/F(p2)−1. In this plot a−1 ≈3.6 GeV. H (q,k) = 1 is not satisfied. This suggests that the scalar function H (q,k) plays an 1 1 important role in the infrared gluodynamics. Finally, a direct test of the relation (7) and thus, as we have seen in (12), of the value of α can be done. For this we plot the quantity Γ F2(p2)G(p2) Fig.3. If all the conditions (12) are satisfied this quantity should be constant in the infrared. We see from Fig.3 that in the infrared (below ≈ 600 MeV) the quantity F2G is not constant, and thus one of the conditions (12) is not verified. We have seen that the conditions α 6= 0 and α +α < 1 are consistent with the limits (6) from the F F G Slavnov-Taylor identity (5). We have also seen (cf. Fig.1,2) that neglecting the vertex is not possible in the infrared, because in this case the ghost SD equation is no longer satisfied by lattice propagators. Thus the only possibility is to admit that H (q,k) plays 1 an important role, and that the relation (7) is not verified. The modified form (13) that takes in account the singularity of H (q,k) should be considered (according to Fig.2), 1 with α < 0 in our parametrisation. Γ 4 25 15 SU3 β=5.75 Vol=324 20 15 10 G G 2F 2F 10 SU2 β=2.3 Vol=324 5 SU2 β=2.3 Vol=484 5 0 0 0 0,5 1 1,5 0 0,5 1 1,5 ap ap Figure 3: Direct test of the relation 2α +α = 0. If the last is true F2G has to be constant in the F G infrared. We see that it is clearly not the case. In these plots a−1 ≈ 1.2 GeV, so the peak is located at ≈600 MeV. 4 Conclusions We have seen that lattice simulations allow a whole momentum range study of Green functions of a non-Abelian gauge theory in Landau gauge. They have been tested by perturbation theory up to NNNLO at large momentum. We have also checked (numerically) that lattice Green functions satisfy the complete ghost Schwinger-Dyson equation (14) for all considered momenta. These tests allow us to conclude that numerical simulation on the lattice give relevant results not only in the ultraviolet domain but also in the infrared one. Our analysis of the Slavnov-Taylor identity showed that the power-law infrared divergence of the ghost propagator is enhanced in the infrared (compared to the free case), and that the gluon propagator must diverge in the infrared. The latter limit is in conflict with most present analytical estimations [3], that support a vanishing gluon propagator in the infrared. In [4] the author replaced H (q,k) by its perturbative expansion, and an interval 1 of values of α was found, including those compatible with (5). Lattice simulations point G to a finite non-vanishing infrared gluon propagator, in conflict with the above analysis of the Slavnov-Taylor identity. Our numerical studies showed that the commonly accepted relation (7) between the infraredexponents is not valid, because F2Gis infraredsuppressed, andhence 2α +α > F G 0. This statement is supported by the fact that lattice propagators do not match the reduced SD equation (Fig.2), whereas the complete one is perfectly verified (Fig.1). This speaks in favour of a singularity in the scalar factor H . Note that it does not contradict 1 the non-renormalisation theorem [6] which implies for the renormalisation constant of the ghost-gluon vertex : ZMS = 1. 1 References e [1] P. Boucaud et al., Phys. Rev. D 72 (2005) 114503 [arXiv:hep-lat/0506031]. 5 [2] P. Boucaud et al., to appear in JHEP [arXiv:hep-lat/0507005]. [3] C. Lerche and L. von Smekal, Phys. Rev. D 65 (2002) 125006[arXiv:hep-ph/0202194]. D. Zwanziger, Phys. Rev. D 67 (2003) 105001 [arXiv:hep-th/0206053]. C. S. Fischer and H. Gies, JHEP 0410 (2004) 048 [arXiv:hep-ph/0408089]. [4] J. C. R. Bloch, Few Body Syst. 33 (2003) 111 [arXiv:hep-ph/0303125]. [5] F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. G. Williams, J. M. Zanotti, Phys. Rev. D 64 (2001) 034501 [ arXiv: hep-lat/0101013] [6] J. C. Taylor, Nuclear Physics B Volume 33, Issue 2 , 1 November 1971 A. A. Slavnov, Theor. Math. Phys. 10 (1972) 99 [Teor. Mat. Fiz. 10 (1972) 153]. [7] P. Boucaud et al., arXiv:hep-ph/0507104. [8] K. G. Chetyrkin and A. Retey, [arXiv:hep-ph/0007088]. 6